# Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem

- Volume: 48, Issue: 3, page 859-874
- ISSN: 0764-583X

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topBurman, Erik, and Hansbo, Peter. "Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.3 (2014): 859-874. <http://eudml.org/doc/273126>.

@article{Burman2014,

abstract = {We extend our results on fictitious domain methods for Poisson’s problem to the case of incompressible elasticity, or Stokes’ problem. The mesh is not fitted to the domain boundary. Instead boundary conditions are imposed using a stabilized Nitsche type approach. Control of the non-physical degrees of freedom, i.e., those outside the physical domain, is obtained thanks to a ghost penalty term for both velocities and pressures. Both inf-sup stable and stabilized velocity pressure pairs are considered.},

author = {Burman, Erik, Hansbo, Peter},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {finite element methods; stabilized methods; penalty methods; Stokes’ problem; fictitious domain; Stokes' problem},

language = {eng},

number = {3},

pages = {859-874},

publisher = {EDP-Sciences},

title = {Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem},

url = {http://eudml.org/doc/273126},

volume = {48},

year = {2014},

}

TY - JOUR

AU - Burman, Erik

AU - Hansbo, Peter

TI - Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2014

PB - EDP-Sciences

VL - 48

IS - 3

SP - 859

EP - 874

AB - We extend our results on fictitious domain methods for Poisson’s problem to the case of incompressible elasticity, or Stokes’ problem. The mesh is not fitted to the domain boundary. Instead boundary conditions are imposed using a stabilized Nitsche type approach. Control of the non-physical degrees of freedom, i.e., those outside the physical domain, is obtained thanks to a ghost penalty term for both velocities and pressures. Both inf-sup stable and stabilized velocity pressure pairs are considered.

LA - eng

KW - finite element methods; stabilized methods; penalty methods; Stokes’ problem; fictitious domain; Stokes' problem

UR - http://eudml.org/doc/273126

ER -

## References

top- [1] S. Amdouni, K. Mansouri, Y. Renard, M. Arfaoui and M. Moakher, Numerical convergence and stability of mixed formulation with X-FEM cut-off. Eur. J. Comput. Mech.21 (2012) 160–73.
- [2] S. Amdouni, M. Moakher and Y. Renard, A local projection stabilization of fictitious domain method for elliptic boundary value problems. Preprint, hal.archives-ouvertes.fr: hal-00713115 (2012) Zbl1288.65152MR3131863
- [3] Ph. Angot, A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions. C.R. Math. Acad. Sci. Paris348 (2010) 697–702. Zbl1194.35317MR2652501
- [4] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equation. Calcolo21 (1984) 337–344. Zbl0593.76039MR799997
- [5] R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo38 (2001) 173–199. Zbl1008.76036MR1890352
- [6] R. Becker, E. Burman and P. Hansbo, A finite element time relaxation method. C.R. Math. Acad. Sci. Paris349 (2011) 353–356. Zbl1305.76052MR2783334
- [7] R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg.198 (2009) 3352–3360. Zbl1230.74169MR2571349
- [8] R. Becker and P. Hansbo, A simple pressure stabilization method for the Stokes equation. Commun. Numer. Methods Eng.24 (2008) 1421–1430. Zbl1153.76036MR2474694
- [9] M. Bercovier and O. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math.33 (1979) 211–224. Zbl0423.65058MR549450
- [10] S. Bertoluzza, M. Ismail and B. Maury, Analysis of the fully discrete fat boundary method. Numer. Math.118 (2011) 49–77. Zbl1217.65201MR2793902
- [11] D. Boffi, F. Brezzi, L. Demkowicz, R. Durán, R. Falk and M. Fortin, Mixed finite elements, compatibility conditions, and applications. Lectures given at the C.I.M.E. Summer School held in Cetraro 2006, edited by Boffi and Lucia Gastaldi. In vol. 1939 Lect. Notes Math. Springer-Verlag, Berlin (2008). Zbl1182.76895MR2459075
- [12] F. Brezzi and J. Pitkäranta, On the stabilization of finite element approximations of the Stokes equations, in Efficient solutions of elliptic systems (Kiel, 1984), vol. 10 of Notes Numer. Fluid Mech. Vieweg, Braunschweig (1984) 11–19. Zbl0552.76002MR804083
- [13] F. Brezzi and R. Falk, Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal.28 (1991) 581–590. Zbl0731.76042MR1098408
- [14] E. Burman and P. Hansbo, Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method. Appl. Numer. Math. 62 (2012) 328–341. Zbl1316.65099MR2899249
- [15] E. Burman and P. Hansbo, Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Engrg.195 (2006) 2393–2410. Zbl1125.76038MR2207476
- [16] E. Burman, Pressure projection stabilizations for Galerkin approximations of Stokes’ and Darcy’s problem. Numer. Methods Part. Differ. Eqs.24 (2008) 127–143. Zbl1139.76029MR2371351
- [17] E. Burman, Ghost penalty. C.R. Math. Acad. Sci. Paris348 (2010) 1217–1220. Zbl1204.65142MR2738930
- [18] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 2. Functional and Variational Methods. Springer-Verlag, Berlin (1988) Zbl0683.35001MR969367
- [19] C. Dohrmann and P. Bochev, A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids46 (2004) 183–201. Zbl1060.76569MR2079895
- [20] V. Girault, R. Glowinski and T. Pan, A fictitious–domain method with distributed multiplier for the Stokes problem, in Appl. Nonlinear Anal. Kluwer/Plenum, New York (1999) 159–174. Zbl0954.35127MR1727447
- [21] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg.47 (2009) 5537–5552. Zbl1035.65125MR1941489
- [22] J. Haslinger and Y. Renard, A new fictitious domain approach inspired by the extended finite element method. SIAM J. Numer. Anal.191 (2002) 1474–1499. Zbl1205.65322MR2497337
- [23] G. Legrain, N. Moës and A. Huerta, Stability of incompressible formulations enriched with X-FEM. Comput. Methods Appl. Mech. Engrg.197 (2008) 1835–1849. Zbl1194.74426MR2417162
- [24] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36 (1971) 9–15. Zbl0229.65079MR341903

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